I've seen that in order to introduce the Hamiltonian approach to the analysis of a mechanical system, the notion of symplectic manifold and symplectomorphism are fundamental.
Moreover, I've read a lot of references that define the cotangent bundle to be the perfect prototype of a symplectic manifold and hence endow it with the canonical 2-closed form $\omega=\sum_{i=1}^n dq_i\wedge dp_i$.
So my question is, why is the pair $(T^*Q,\omega)$, defined in this way with $Q$ configuration manifold, so important from the point of view of Hamiltonian mechanics and Symplectic Geometry?
I think I don't get even how such a 2-closed form $\omega$ is constructed starting from the local coordinates of the manifold $q_1,...,q_n$ and the projections between the various tangent and cotangent spaces.
So if you can explain to me even that construction or reference me to a simple explanation it would be very helpful.